Question

$$\text { Evaluate exactly: } \sin \left(2 \sin ^{-1} 1\right)$$.

   $$\text { Evaluate exactly: } \sin \left(2 \sin ^{-1} 1\right)$$.

Precalculus

Cynthia Young 2nd Edition

Chapter 6, Problem 129

Instant Answer

Solved by Expert Shamshad Waris

06/29/2020

Step 1

We know that $\sin ^{-1} 1$ is $\frac{\pi}{2}$, so we can substitute this into the expression. Show more…

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$$\text { Evaluate exactly: } \sin \left(2 \sin ^{-1} 1\right)$$.

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Key Concepts

Inverse Sine Function

The inverse sine function, denoted as sin?¹ or arcsin, is the function that returns the angle whose sine is a given number. It has a principal range of [-?/2, ?/2] to ensure that each value in its domain corresponds to exactly one angle, making the function well-defined. This concept is crucial in determining angles from known sine values, which then can be used in further trigonometric operations.

Double Angle Identity

The double angle identity for sine states that sin(2?) = 2sin?cos?. This identity is essential in simplifying expressions that involve the sine of twice an angle by relating it to the sine and cosine of the original angle. It provides a method to express trigonometric functions of multiple angles in terms of single-angle functions, which can often be evaluated exactly using known trigonometric values.

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